This notion was introduced by Willett and Yu in [1]} - to deal with a more general version of Example REF  below. For what we will do in this paper, we will consider situation that either \(\mathcal {X}\)  or \(\mathcal {Y}\)  is a constant family - i.e \(X_{m}\)  (resp. \(Y_{m}\) ) are all the same metric space. This is more general than the situation of [1]}, as it captures certain direct limits or elementary convergences in the space of marked groups (see for instance [3]}).
